# Ch 6  Inverse Circular Functions and Trigonometric Equations

## Ch 6.1  Inverse Circular Functions

Inverse Functions

Animation: Illustrate why a function must be one-to-one to have an inverse function

Introduction to Inverse Sine, Inverse Cosine, and Inverse Tangent

Introduction to Inverse Cosecant, Inverse Secant, and Inverse Cotangent :

The Domain of y=sin(x) is [ , ] and the Range is [–1, 1]
The Domain of y=sin-1(x) is [–1, 1] and the Range is [ , ]
The Domain of y=cos(x) is [0, π] and the Range is [–1, 1]
The Domain of y=cos-1(x) is [–1, 1] and the Range is [0, π]
The Domain of y=tan(x) is [ , ] and the Range is [–∞ ∞]
The Domain of y=tan-1(x) is [–∞, ∞] and the Range is [ , ]
Finding the Exact Value of y=arcsin( )
Finding the Exact Value of y=arccos( )
Finding the Exact Value of y=tan-1(–1)

Finding the Exact Inverse Function Values Involving Inverse Sine, Inverse Cosine, and Inverse Tangent :

Evaluating y=sin-1( )
Evaluating y=cos-1(0)
Evaluating y=arctan(–1)

Evaluating y=sin-1(sin( ))
Evaluating y=sin-1(cos( ))
Evaluating y=tan-1(sin( ))

Finding the Exact Inverse Function Values Involving Inverse Cosecant, Inverse Secant, and Inverse Cotangent  :

Evaluating csc-1( ) in Degree and Radian

Evaluating arcsec(–2) in Degree and Radian

Evaluating sin-1( )
Evaluating sec-1(2)
Evaluating csc-1( )
Evaluating cot-1(–1)

Determining the Exact Value without a Calculator (Part 1 of the video) :
arcsec(2)
arccsc( )
arccot(–1)

Finding a Inverse Cotangent Value in Degrees and Radians Using a Calculator :

arccot(–3.6)

Finding the Exact Function Values Involving Inverse Sine, Inverse Cosine, and Inverse Tangent :

sin(arccos( ))
cos(arctan( ))
sin(sin-1( )+tan-1(√–3))
Finding the Angle the Ladder Makes with the Ground
Finding the Maximum Angle of Elevation to Maximize a Shot Putter Distance

Evaluating sin(sin-1( ))
Evaluating cos(cos-1( ))
Evaluating tan(cos-1( ))

Evaluating tan(sin-1( ))

Evaluating sin(tan-1(–7))

Evaluating sin(tan -1( )) and Assume u>0

Finding an Exact Sine Function Value Containing an Inverse Cosine – Double Angle :
sin(2arccos( ))

Finding the Exact Function Values Involving Inverse Cosecant, Inverse Secant, and Inverse Cotangent  Without a Calculator (Part 2 of the video) :
csc(arccot(u))
cos(sec-1( ))
sec(arccot( ))

## Ch 6.2  Trigonometric Equations I

Solving a Trigonometric Equation by Linear Method :

Solving Each Equation on the Interval [0, 2π) and then Over All Radian Solutions
2sinθ–1=0
2cosθ+√2=0
√3tanθ–1=0
4cosθ–6=cosθ

Solving sin(x)+ =0 on the Interval [0, 360°)

Solving 2cos(x)sin(x)=sin(x) on the Interval [0, 360°)

Solving 3tan2(x)–1=0 on the Interval [0, 2π)

Solving 3=20sin(x–3)+1 on the Interval [0, 2π) (Part 1 of the video)

Solving  on the Interval [0, 2π) (Part 1 of the video)

Example: Solving Trigonometric Equation: sin(x)=cos(x)

Solving a Trigonometric Equation by Factoring :

Solving tan2θ–1=0 on the Interval [0, 2π)
Solving 2cos2θ–√3cosθ=0 on the Interval [0, 2π)
Solving 2sin2θ=–3sinθ–1 on the Interval [0, 2π)

Solving 2cos2(x)–sin(x)=1 on the Interval [0, 2π)

Solving cos2(x)–cos(x)–2=0 on the Interval [0, 2π)

Solving a Trigonometric Equation by Trigonometric Identities :

Solving cos2(x)–sin2(x)=  on the Interval [0, 360°)
Solving tanθ+√3=secθ on the Interval [0, 2π)

Solving sec2(x)–2tan(x)=4 on the Interval [0, 2π) (Part 2 of the video)

Solving a Trigonometric Equation Using the Calculator :

Solving sin(x)–0.32=0 on the Interval [0, 2π)

Solving cos(x)+0.85=0 on the Interval [0, 2π)

Solving Applications Problems :

The Function T(x)=19sin( x – )+53  Modeling the Average Monthly Temperature of Water in a Mountain Stream
The Function S(x)=1600cos( x+ )+5100 Modeling the Average Monthly Sales in the Month x

Determining the Height of an Object Using a Trigonometric Equation

## Ch 6.3  Trigonometric Equations II

Solving a Trigonometric Equation by Double Angle :

Solving cos(2x)+sin2(x)–3cos(x)=1 on the Interval [0, 360°) (Part 3 of the video)

Solving cos(2θ)–cos(θ)=0 on the Interval [0, 2π)
Solving sin(θ)–sin(2θ)–=0 on the Interval [0, 2π)
http://www.youtube.com/watch?v=8FRly0POPD8 (Part 3 and 4 of the video)

Solving sin(2x)=cos(2x)+1 on the Interval [0, 2π) (Part 3 of the video)

Solving cos(2x)=cos(x) on the Interval [0, 2π)

Solving sin(2x)=2cos2(x) on the Interval [0, 2π)

Solving a Trigonometric Equation by Half Angle (Part 1 of the video):
sin(x/2)=√2–sin(x/2)

Solving a Multiple-Angle–Trigonometric Equation by Single Angle :

Solving 4cos(4x)=2 on the Interval [0, 2π)

Solving 5sin(3x)=2 on the Interval [0, 2π)

Solving 2cos(3x)–√3=0 on the Interval [0, 360°) (Part 2 of the video)

## Ch 6.4  Equations Involving Inverse Trigonometric Functions

Solving the Equation for Secant x :
√5+2sec(3x)=y for x:

Solving a Trigonometric Equation with an Inverse Trig Function :

4arctan(x)=π;
cos-1(x)=sin-1(3/7)
(Part 2 and 3 of the video)

3/4cos-1(y/3)=(π/2)

4/7cos-1(x/4)=π